Quantum Information for Quantum Cats (18)

Quantum Circuit : One and Two Qubit Operators

From a flashy subject of teleportation, we come back to reality and deal with down-to-earth subject of producing a calculator.

Usually a calculator circuit is made up from elementary circuits. The problem becomes that of exausting a complete set of elementary circuits whose combination can produce universal calculation. As you might easily check, addition, subtraction, multiplification and division in combination make up the complete set. In terms of bit operation, these reduce to a set consisting of NOT, XOR, and AND. Can we prepare quantum equivalent of this complete set?

We use Up-Down coding from now on, and adopt identifications | 0 >= | u >, | 1 > = | d >. Consider three one qubit operations

I : | 0 > to | 0 >, | 1 > to | 1 > X : | 0 > to | 1 >, | 1 > to | 0 > Z : | 0 > to | 0 >, | 1 > to -| 1 >

The first one I is an identity. The second one X is nothing but .not on a qubit. The third one adds a relative sign whose meaning is unclear at this point. We next consider transformations on two qubit states. Beacuse of the requirement of "unitarity" of quantum transformation, which is a reflection of fundamental symmetry of nature, there are certain limitations to what we can do. Reversibility is one such requiremennt that demands every transfromation must have its reverse as a legitimate transformation. Naturally, input and output should have same number of qubits. One example of reversible two qubit operation is

CN :

| 0 0 > to | 0 0 >, | 0 1 > to | 0 1 > | 1 0 > to | 1 1 >, | 1 1 > to | 1 0 >

The higher bit just gets through. The behavior of the lower bit depends on the value of the higher bit: When the higher bit is 0, the lower bit is also unchnaged, but hwne higher bit is 1, the value of the lower bit gets reversed, which is to say, gets the effet of .not. operation. We can formally express this as

This operation is called "Control-Not", and .xor. is obtained from it as CN | 0 y > = .xor. | 0 y >. Now hat we have established quantum operation ofr .not and of .xor., we want to have .and. to complete the universal set of operations. However, it is proven that one cannot produce .and. from two qubit operation.